In contrast to the power function, where the variable is in the base, the exponential function the variable in the exponent.
The functional equation of an exponential function
An exponential function with the base is a real function of the form:
means that a (called: «the base») is greater than 0 and must not be 1 at the same time. The variable x is in the exponent.
Because the variable is in the exponent, this function is called an “exponential function”.
The exponential function with a prefactor b
An exponential function can also have a prefactor b, this factor is a real number, but it should not be 0. Otherwise the total result of the function would eventually be 0.
The functional equation then looks like this:
The following are a few examples of what a function term of an exponential function with a prefactor might look like:
- here a=2 and b=-3
- here a=7 and b=1.5
- , this term is not that of an exponential function because the variable is not in the exponent. It is the function term of a power function
The natural exponential function and Euler’s number
The is particularly important for the inverse function and also the differentiability and integrability calculation Euler’s number e.
The Euler number is the limit value:
e is an irrational number. You can also write this as a decimal fraction. It is infinite but not periodic and starts at 2.71828…
The associated exponential function of e is called the e-function or natural exponential function.
This number is particularly important in the case of exponential growth, eg the growth of bacteria, or also exponential decrease processes.
The natural exponential function has the form . The number e is in the base instead of the coefficient.
The inverse of the exponential function
You can also convert any exponential function into a natural exponential function, the so-called «e-function» or «Euler’s number». This natural exponential function then has the base e. e is Euler’s number.
You can also use this relationship to determine the derivative. The natural logarithm function, ln function, is the inverse of the e function. This means:
Here you can see: If you reflect the e-function at the bisecting line (x=y), you get the ln-function.
The derivative of the exponential function
The derivative of the exponential function is:
The antiderivative of the exponential function
The antiderivative or the integral F(x) of the exponential function is:
Our tip for you
I would recommend that you read the Exponential Growth article thoroughly and do the sample tasks yourself. There you will find special application examples for the theory learned above and see that this topic is also very important in everyday life. This will allow you to internalize the knowledge you have learned!